A Statistical Analysis of the Higgs Boson

Abstract: The writing begins with basic statistics, null hypothesis, signal-free datasets and

move on to more advanced statistical analysis. The final report scrutinizes the standard

model H in pp collisions at √s = 13 TeV using pseudo-experiment training datasets to

optimize event selection.

Shouyang Michael Wang

University of Washington

Physics 434 Capstone Project

Advisor: Dr. Miguel Morales

December 2019

2 
 
 
 
Contents 
Background ................................................................................................................................................... 3 
Statistics ........................................................................................................................................................ 5 
Normal Distribution ...................................................................................................................................... 5 
χ2-distribution ............................................................................................................................................... 6 
Poisson distribution ...................................................................................................................................... 7 
Central Limit Theorem .................................................................................................................................. 8 
Multidimensional Gaussian ........................................................................................................................... 9 
Higgs Bosons & QCD Parameters ................................................................................................................ 11 
DATA EXPLORATION.................................................................................................................................... 12 
TRANSVERSE MOMENTUM(PT) (XY PLANE)(UNIT: GEV*C-1 OR GEV) ........................................................ 13 
PSEUDORAPIDITY ETA (η)............................................................................................................................ 14 
PHI (Φ) ......................................................................................................................................................... 15 
INVARIANT MASS (M)(UNIT: GEV*C-2 OR GEV) ......................................................................................... 16 
EE2(E2) AND EE3(E3): ENERGY CORRELATION FUNCTIONS ........................................................................ 18 
D2(D2) JET SHAPE FUNCTION, USED IN TAGGING ...................................................................................... 20 
ANGULARITY ............................................................................................................................................... 21 
N-SUBJETTINESS(𝜏N); JET SHAPE FUNCTION, USED IN TAGGING ............................................................... 22 
KTDELTAR(KT ΔR) – KT CLUSTERING ........................................................................................................... 24 
Higgs Sensitivity .......................................................................................................................................... 25 
Conclusions ................................................................................................................................................. 27 
Bibliography ................................................................................................................................................ 28 
 
 
 
 
 
 

Background

The Standard Model

The Standard Model of particle physics describes all known fundamental particles that are

the building blocks to everything in the universe.

The Higgs mechanism, a mechanism theorized to give rise to the masses of all the

subatomic elements, is essential for the Standard Model to work mathematically - all

particles that mediate forces are massless. The mechanism successfully explained how

particles obtain their mass through the Higgs field and theorized the Higgs boson’s

existence.

Higgs boson (H) is one of the elementary particles in the Standard Model. The particle is

extremely unstable, meaning that it would decay into other particles almost immediately.

Higgs boson has been verified experimentally using the Large Hadron Collider at the

European Organization for Nuclear Research(CERN).

Large Hadron Collider

The Large Hadron Collider is a circular high-energy particle accelerator and collider. It

consists of a ring of “superconducting magnets,” along with several accelerating structures.

Inside the ring, two beams containing protons or lead ions would travel in opposite

directions to collide at speed close to the speed of light. Particle detectors are placed

around the collision site to record and analyze the subatomic particles that arise from the

collision. The LHC is engineered to operate at a maximum collision energy of 14 TeV,

meaning that each traveling beam would carry a power of 7 TeV. However, in most

operations, the collision energy is controlled under 14 TeV to “optimize the delivery of

particle collisions” (CERN).

Furthermore, one of the particle detectors is named ATLAS(A Toroidal LHC ApparatuS). It

is used to detect a wide range of subatomic particles, including the Higgs boson.

(Source: CERN)

Collision Energy

In the context of proton-proton(pp) collisions, √𝑠 indicates the total center-of-mass energy

of the colliding beams.

In the laboratory frame, the center-of-mass energy is calculated as the following:

We will observe and analyze an inclusive search for the standard model Higgs boson in

typical pp collisions at √𝑠 = 13 TeV at the LHC using simulated data. In other words, the

energy for one beam of protons is 6.5 TeV, which would result in the total energy of √𝑠 =13

TeV when two beams collide.

Jets

The term refers to a large number of detectable particles flying roughly in the same

direction. The process is called clustering.

The decay of Higgs boson

The Higgs boson is extremely unstable, and it would decay into other particles almost

immediately. The standard model of particle physics theorized that the Higgs could decay

into other particles like quarks. There are many “flavors” when it comes to quarks. The net

electric charge of the byproducts of the decay has to be zero since the Higgs boson has a

charge of zero. Quarks have electric charges, meaning that the quarks created from the

decay have to emerge in pairs, such that the charge from one quark and one antiquark

would cancel each other out, resulting in a net charge of zero. Very often, physicists find

that the Higgs boson would decay into a pair of bottom quarks.

Quantum Chromodynamics(QCD)

During a proton-proton(pp) collision, almost all detected events are due to the strong

interaction(QCD). Due to the Higgs boson’s unstable nature, the pp collision would lead to a

pair of “particle jets originating from b quarks’ fragmentation. (ATLAS)” It is incredibly

challenging to distinguish an actual Higgs signal from the” unwanted” b-quark pairs noise

due to the QCD background.

(Source: Particle Physics Theory, University of Glasgow)

Statistics

Normal Distribution

In physics, σ refers to a probability. The sigma implicitly refers to the standard normal

distribution (a Gaussian with mean zero and a standard deviation of 1).

Integrals of the standard normal distribution give probabilities. The normal distribution is

a symmetric distribution where most of the observations cluster around the center to form

a bell shape. A Gaussian would remain a Gaussian after summation, multiplication, or

Fourier transform.

The z-score is positive if the value lies above the mean and negative if it lies below the

mean. Notice that it is possible to get a “negative value” when integrating from a negative

number to another number.

For instance, if we try to integrate a function from negative infinity to a number, it’s

possible to get a negative area. It does not make any sense to have a negative probability.

The probability from -10 to 0 should have the same probability from 0 to 10 for a Gaussian

distribution since the distribution is symmetrical.

To illustrate this, we try to integrate the equation for the normal distribution using two

different sets of limits.

Notice that the two equations generate the same probability. In other words, the

area(probability) under the curve from -10 to 0 is the same as the area from 0 to 10.

χ2-distribution

Now we explore some other continuous analytic distributions. The χ2-distribution (or Chi-

squared) with k degrees of freedom is the distribution of a sum of squares of k independent

standard normal random variables. Now we create a χ2-distribution object with the

parameter value b = 2, a special case of Gamma distribution.

Assuming that we have signal-free data that follows the χ2-distribution, and we have a

measurement for which we need to determine the σ. The test statistic is measured to be

7.1. We propose a statistical question: What is the probability that the background

produces a signal that is less than the measurement we made? More specifically, given the

test statistics and if we know the distribution has 3 degrees of freedom, what is the area to

the left of the test statistic? First, we try to approach the question mathematically. The

formula for the cumulative distribution function of the chi-square distribution is given by

Poisson distribution

Now we move on to non-continuous distributions. We will be looking at the Poisson

distribution.

Notice that

1) all of the variables are discrete.

2) The graphs have a lower bond at zero(For instance, the data can’t go lower than 0), but

they don’t have an upper limit.

3)The graph with a low mean(lambda) is highly skewed. 4) the distribution becomes more

bell-like as the mean is getting large.

5) The shape isn’t very symmetric like the normal distribution.

6) The mean, lambda, is a constant number throughout the experiment.

We propose a statistical question: Given that lambda = 3, what is the probability of the

background producing a precisely 4? It can be calculated as the following:

Because the distributions are discrete, so are the probabilities and σ. A discrete number has

countable values. Since the distribution is discrete, it makes sense to talk about probability

at an exact value like P(x=4) shown above. The discrete nature is more pragmatic when it

comes to experiments that expect a specified number of events to take place.

While the results are discrete, the parameters of the distributions are not. For example, the

mean of a Poisson distribution can be 9.2. The mean is essentially the average of a set of

numbers. It is an arithmetic number obtained from experimental data. It does not have to

be a whole number. However, since the distribution is typically used to reflect the

probability of a number of events occurring in a fixed interval, it does not make much sense

to say that an event happened 1.5 times. An event can happen on average 1.5 times.

Typically the x-axis represents a given interval of time(e.g., how many times an event

happens at hours 1, 2,3,4, etc.). Just like it’s either head or tail, it can’t be somewhere in

between.

Central Limit Theorem

Furthermore, the sum of a Poisson with itself still has Poisson distribution. To prove this,

we will use the moment generating function of the Poisson distribution. It is given by

For a linear combination,

Hence,

LaTex code:

From a conceptual point of view, the Poisson distribution is the number of counts in a fixed

interval of time. When we conv() or sum a Poisson distribution with itself N(in this case,

five) times, it is the same as saying that we multiply the interval by N, or N times longer

than the original interval. Consequently, the mean is also multiplied by N. The two methods

(sum N times or multiply interval and mean by N)would get the same result. A Poisson

distribution with a rescaled interval and mean is still a Poisson distribution.

A limiting form of the Poisson distribution is the Gaussian distribution. The central limit

theorem states that when independent random variables are summed or averaged enough

trials, their normalized sum or average tends toward a normal distribution. Poisson is

becoming more symmetrical and bell-like as it is averaging over time. Because of the

central limit theorem, Poisson tends to normal as its mean becomes sufficiently large.

Multidimensional Gaussian

Higgs Bosons & QCD Parameters

DATA EXPLORATION

The Higgs bosons are produced with large transverse momentum (pT) and decay to a

bottom quark-antiquark pair. The Higgs candidates could be reconstructed as large-radius

jets using Calorimeters. Due to large QCD background contamination, the direct 5-sigma

observation of this Higgs channel is not accomplished yet[Phys. Rev. Lett. 120, 071802

(2018)]. We will analyze the data using MATLAB. See lab5.m for the complete code. The

signal dataset is labeled as “Higgs,” and the background dataset is labeled as “QCD.” First,

we import the data. In the data, the transverse momentum ranges from 250 GeV to 500

GeV.

As shown above, there are two datasets. The first dataset is the Higgs signal, and the second

dataset is the QCD background. Each dataset contains 14 rows and 100000 columns. 14

rows represent the 14 parameters we mentioned previously. They are ‘pt', 'eta', 'phi',

'mass', 'ee2', 'ee3', 'd2', 'angularity', 't1', 't2', 't3', 't21', 't32' , and 'KtDeltaR'. For each

parameter, we have recorded 100000 datapoints.

TRANSVERSE MOMENTUM(PT) (XY PLANE)(UNIT: GEV*C-1 OR

GEV)

The component of momentum in the transverse plane (in other words, perpendicular to

the beamline.) The parameter is always associated with the events that happened at the

vertex, which is a fairly clean sample.

Notice that the x-axis goes from 250 to 500. This agrees with the dataset we use where the

transverse momentum ranges from 250 GeV to 500 GeV. One peculiar thing to notice is that

there is a noticeable gap at the very end of the plot — the Higgs datapoints peak around

485 GeV.

PSEUDORAPIDITY ETA (η)

Eta describes the angle of a particle relative to the beam axis(z-axis). θ and η are

convertible.

PHI (Φ)

INVARIANT MASS (M)(UNIT: GEV*C-2 OR GEV)

When a Higgs boson particle decays into other particles, its mass before the decay can be

calculated “from the energies and momenta of the decay products”(ATLAS). The mass is an

invariant quantity, which is the same for all observers in all reference frames.

We will plot the Higgs and QCD background mass and compare the two plots.

The difference between the Higgs and QCD data is fairly noticeable. Here, we are looking at

the “irregularities” of the plot, which would give us information about the signal.

EE2(E2) AND EE3(E3): ENERGY CORRELATION FUNCTIONS

More on the Energy Correlation Functions

Energy Correlations Functions can be used as an inquiry for the jet substructure, meaning

that we can use the functions to identify the N-prong substructure without a subject finding

procedure.

We can use ECFs to discriminate between QCD jets and Higgs Bosons.

A noticeable difference between the signal and background occurs when 2-point ECF is

greater than 0.11.

D2(D2) JET SHAPE FUNCTION, USED IN TAGGING

Note: The QCD background covers the Higgs data. The two datasets share similar

distributions.

ANGULARITY

Notes:

“Angularities are a family of observables that are sensitive to the degree of symmetry in the

energy flow inside a jet…..The measurement is aimed primarily at testing QCD, which

makes predictions for the shape of the angularity distribution in jets where the small-angle

approximation is valid…….Angularity is largely uncorrelated with all the other

observables.” (source: https://arxiv.org/pdf/1206.5369.pdf)

The Higgs and QCD data follow similar distributions.

N-SUBJETTINESS(𝜏N); JET SHAPE FUNCTION, USED IN TAGGING